In applications such as radar, sonar, data communications, time series analysis, and array processing, an object is to determine whether a specific signal is present in a series of N measured data samples (which can be represented as Z=[z(0), z(1), . . . , z(N−1)]r) that contain unknown interference and noise. Based on these data samples, a decision must be made between two possible hypotheses, viz., the null hypothesis H0 in which the data consists of interference only, and the alternate hypothesis H1 in which the signal is present in the data as well. These two hypotheses are exemplified via the mathematical representation of the measured data samples:z=αs+n  (1)where s is the signal vector, with α is its associated complex amplitude, and n is the noise plus interference. Under hypothesis H0, the signal amplitude is α=0, whereas under hypothesis H1, the signal amplitude is α≠0. The covariance of the noise plus interference is R, which is employed in Matched Subspace Detectors (MSDs) to effectively suppress the noise and interference to enable reliable detection performance (as a function of the signal-to-noise ratio (SNR) and the separability of the signal and interference). (See, for example, L. L. Scharf and B. Friedlander, “Matched subspace detectors,” IEEE Trans. Signal Processing, Vol. 42, No. 8, pp. 2146–2157, August 1994.) In practice, however, R is not known and must, therefore, be estimated. Hence, the detector performance is also highly dependent upon the accuracy of the covariance matrix estimate {tilde over (R)}. MSDs that use the estimated covariance matrix are known as Adaptive Subspace Detectors (ASDs) because they adapt to the measured data.
An Adaptive Coherence Estimate (ACE) detector, which is also known as an Adaptive Cosine Detector, is one such ASD in which the specific form of the desired signal is known (as opposed to detectors that test for the presence of any signal that lies within the signal subspace), but the power level of the noise and interference is unknown (see, e.g., L. L. Scharf and L. T. McWhorter, “Adaptive matched subspace detectors and adaptive coherence estimators,” Proc. 30th Asilomar Conf. on Signals, Systems, and Computers, Vol. 1, pp. 1114–1117, Nov. 3–6, 1996; L. T. McWhorter, L. L. Scharf, and L. J. Griffiths, “Adaptive coherence estimation for radar signal processing,” Proc. 30th Asilomar Conf. on Signals, Systems, and Computers, Vol. 1, pp. 536–540, Nov. 3–6, 1996; and S. Kraut, L. L. Scharf, and L. T. McWhorter, “Adaptive subspace detectors,” IEEE Trans. Signal Processing, Vol. 49, No. 1, January 2001]). For each range index k the ACE takes the form
                              ACE          ⁡                      (            k            )                          =                                                                                            s                  H                                ⁢                                                      R                    ~                                                        -                    1                                                  ⁢                                  z                  k                                                                    2                                              (                                                s                  H                                ⁢                                                      R                    ~                                                        -                    1                                                  ⁢                s                            )                        ⁢                          (                                                z                  k                  H                                ⁢                                                      R                    ~                                                        -                    1                                                  ⁢                                  z                  k                                            )                                                          (        2        )            wherein s, R and z are as defined above, and H denotes the Hermitian matrix or complex conjugate transpose. The resulting ACE value for a given data vector zk is then compared with a predetermined threshold to achieve a desired probability of false alarm. The ACE is bounded between 0 and 1 and effectively determines a measure of coherence of the cell-under-test (CUT) with the desired steering vector (that models a target return signal from the corresponding spatial direction and Doppler frequency).